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Properties

Label 119.5.c.a
Level $119$
Weight $5$
Character orbit 119.c
Analytic conductor $12.301$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [119,5,Mod(69,119)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("119.69"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(119, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 119 = 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 119.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.3010256070\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 6 q^{2} + 378 q^{4} - 130 q^{7} - 6 q^{8} - 1232 q^{9} + 192 q^{11} - 294 q^{14} - 128 q^{15} + 3178 q^{16} + 682 q^{18} + 312 q^{21} + 336 q^{22} + 876 q^{23} - 3800 q^{25} - 2426 q^{28} - 3372 q^{29}+ \cdots - 19620 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1 −7.78982 − 5.91608i 44.6813 27.4318i 46.0852i −20.8883 + 44.3247i −223.422 46.0000 − 213.689i
69.2 −7.78982 5.91608i 44.6813 − 27.4318i − 46.0852i −20.8883 − 44.3247i −223.422 46.0000 213.689i
69.3 −6.64373 − 7.69943i 28.1392 − 14.0269i 51.1530i 39.4736 − 29.0316i −80.6497 21.7188 93.1907i
69.4 −6.64373 7.69943i 28.1392 14.0269i − 51.1530i 39.4736 + 29.0316i −80.6497 21.7188 − 93.1907i
69.5 −6.61881 − 16.9193i 27.8087 − 36.7970i 111.986i −48.9549 − 2.10218i −78.1594 −205.263 243.552i
69.6 −6.61881 16.9193i 27.8087 36.7970i − 111.986i −48.9549 + 2.10218i −78.1594 −205.263 − 243.552i
69.7 −6.38864 − 13.7676i 24.8147 24.8373i 87.9561i 44.9393 − 19.5310i −56.3142 −108.546 − 158.676i
69.8 −6.38864 13.7676i 24.8147 − 24.8373i − 87.9561i 44.9393 + 19.5310i −56.3142 −108.546 158.676i
69.9 −5.87245 − 3.58586i 18.4857 − 21.1360i 21.0578i −14.0460 + 46.9437i −14.5972 68.1416 124.120i
69.10 −5.87245 3.58586i 18.4857 21.1360i − 21.0578i −14.0460 − 46.9437i −14.5972 68.1416 − 124.120i
69.11 −4.75861 − 9.07832i 6.64435 26.7001i 43.2002i −43.1291 − 23.2568i 44.5199 −1.41596 − 127.056i
69.12 −4.75861 9.07832i 6.64435 − 26.7001i − 43.2002i −43.1291 + 23.2568i 44.5199 −1.41596 127.056i
69.13 −3.34181 − 7.44795i −4.83229 − 19.1327i 24.8897i 43.5341 + 22.4897i 69.6176 25.5280 63.9380i
69.14 −3.34181 7.44795i −4.83229 19.1327i − 24.8897i 43.5341 − 22.4897i 69.6176 25.5280 − 63.9380i
69.15 −3.33024 − 2.05392i −4.90951 − 41.2315i 6.84005i 1.11842 − 48.9872i 69.6337 76.7814 137.311i
69.16 −3.33024 2.05392i −4.90951 41.2315i − 6.84005i 1.11842 + 48.9872i 69.6337 76.7814 − 137.311i
69.17 −2.77571 − 16.1417i −8.29545 4.35853i 44.8046i 11.8075 + 47.5561i 67.4371 −179.554 − 12.0980i
69.18 −2.77571 16.1417i −8.29545 − 4.35853i − 44.8046i 11.8075 − 47.5561i 67.4371 −179.554 12.0980i
69.19 −1.58622 − 10.7764i −13.4839 0.623247i 17.0937i −46.5664 − 15.2503i 46.7681 −35.1300 − 0.988609i
69.20 −1.58622 10.7764i −13.4839 − 0.623247i − 17.0937i −46.5664 + 15.2503i 46.7681 −35.1300 0.988609i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.44
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 119.5.c.a ✓ 44
7.b odd 2 1 inner 119.5.c.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.5.c.a ✓ 44 1.a even 1 1 trivial
119.5.c.a ✓ 44 7.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(119, [\chi])\).